Question: Solve for $x$ and $y$ using elimination. $\begin{align*}9x-6y &= 3 \\ 6x+7y &= -5\end{align*}$
Answer: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $3$ $\begin{align*}-18x+12y &= -6\\ 18x+21y &= -15\end{align*}$ Add the top and bottom equations. $33y = -21$ Divide both sides by $33$ and reduce as necessary. $y = -\dfrac{7}{11}$ Substitute $-\dfrac{7}{11}$ for $y$ in the top equation. $9x-6( -\dfrac{7}{11}) = 3$ $9x+\dfrac{42}{11} = 3$ $9x = -\dfrac{9}{11}$ $x = -\dfrac{1}{11}$ The solution is $\enspace x = -\dfrac{1}{11}, \enspace y = -\dfrac{7}{11}$.